

A279767


Numbers n such that n and n + 2 have the same prime signature.


2



3, 5, 11, 17, 18, 29, 33, 41, 50, 54, 55, 59, 71, 85, 91, 93, 101, 107, 137, 141, 143, 149, 159, 179, 183, 185, 191, 197, 201, 203, 213, 215, 217, 219, 227, 235, 239, 242, 247, 248, 265, 269, 281, 299, 301, 303, 306, 311, 319, 321, 327, 339, 340, 347, 348, 391, 393, 411, 413
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OFFSET

1,1


COMMENTS

The sequence contains some terms such that n and n + 2k (1 < k) have the same prime signature. For some terms where n and n + 2k share the same prime signature this means that every alternate element between, and including n and n + 2k have the same prime signature. The first such example is where a(41951)=402677, a(41953)=402679, and a(41955)=402681, share the same prime signature {1, 1}. Also the remaining alternate terms excluding endpoints share the same prime signature. Using the above example a(41952)=402678 and a(41954)=402680, share the prime signature {1,1,3}.  Torlach Rush, Feb 25 2018


LINKS

Michel Marcus, Table of n, a(n) for n = 1..5585
Index to sequences related to prime signature


EXAMPLE

18 is a term because 18 = 2 * 3^2 and 18 + 2 = 20 = 2^2 * 5.
19 is not a term because it is prime and 21 is the product of two primes, so the prime signatures are different.


MATHEMATICA

primeSignature[n_] := Sort[Transpose[FactorInteger[n]][[2]]]; Select[ Range[2, 1000], primeSignature[#] == primeSignature[# + 2] &] (* Adapted from A052213 *)


PROG

(PARI) isok(n) = vecsort(factor(n)[, 2]) == vecsort(factor(n+2)[, 2]); \\ Michel Marcus, Feb 25 2018


CROSSREFS

Cf. A001359, A052213, A052214.
Sequence in context: A323582 A088328 A102643 * A125631 A045408 A092740
Adjacent sequences: A279764 A279765 A279766 * A279768 A279769 A279770


KEYWORD

nonn,easy


AUTHOR

Altug Alkan, Dec 18 2016


STATUS

approved



